Invariant Directional Orderings

Abstract

Statistical concepts of order permeate the theory and practice of statistics. The present paper is concerned with a large class of directional orderings of univariate distributions. (What do we mean by saying that a random variable $Y$ is larger than another random variable $X$?) Attention is restricted to preorders that are invariant under monotone transformations; this includes orderings such as monotone likelihood ratio, hazard ordering, and stochastic ordering. Simple characterizations of these orderings are obtained in terms of a maximal invariant. It is shown how such invariant preorderings can be used to generate concepts of $Y_2$ being further to the right of $X_2$ than $Y_1$ is of $X_1$.